The third chapter of A History of Western Philosophy provides a critical exposition of the philosophy of Pythagoras, considered by Russell to be "intellectually one of the most important men that ever lived, both when he was wise and when he was unwise." Following the organizational scheme employed frequently throughout the History, Russell begins by connecting the main episodes in the philosopher's life to the cultural environment from which he emerged. For the remainder of the chapter he turns to the different aspects of Pythagorean thought, considering in turn its mystical components, the underlying metaphysical and religious assumptions and his ethico-political project. This exposition allows Russell on several occasions to make some agreeable theoretical digressions. For example, he etymologizes the word 'theory' and, on establishing its association with the notion of 'contemplation', considers briefly the relationship between the contemplative, the practical and the useful. Russell regards Pythagorean wisdom as the product of two aspects that are "not so separate as they seem to a modern mind": the religious and the purely mathematical.

It is to the latter side of Pythagoras that Russell directs his attention in the final pages of chapter three, emphasizing that he was the first philosopher to perceive not only the 'importance of numbers' but also the overall significance of mathematics. Indeed, 'Mathematics, in the sense of demonstrative deductive argument, begins with him'. In this section Russell does not restrict himself to a formal presentation of Pythagoras' Theorem. He is more inclined to ponder the profound consequences of its discovery. Not the least important effect of the Theorem was that it revealed a new and important problem. 'Unfortunately for Pythagoras,' Russell writes, 'his theorem led at once to the discovery of incommensurables, which appeared to disprove his whole philosophy.' In turn, the failure of Greek arithmetic to confront the puzzle of incommensurables provided a basis for the distinction between geometry and arithmetic which characterized the history of all subsequent mathematical thought, at least until Descartes.

For Russell, a clear illustration of this distinction was contained in Euclid's deductive system. 'Euclid, in Book II, proves geometrically many things which we should naturally prove by algebra, such as (a + b) = a + 2ab + b. It was because of the difficulty about incommensurables that he considered this course necessary'. These reflections prefigure a further theoretical excursus, of astounding density and clarity. It opens with the following proposition: 'The influence of geometry upon philosophy and scientific method has been profound', and it continues through the last three paragraphs of the chapter. The manuscript of the History preserved at the Russell Archives, and discovered on 22 March 2001, however, shows some extraordinarily interesting and important variations from the printed version. Specifically, instead of the last paragraph as it appears in the book, we find nine other paragraphs (i.e. six pages) in which Russell seems to explore further 'The Influence of Geometry on Philosophy'--his title for the above-mentioned excursus, written and underlined at the top in customary fashion two pages earlier. Nowhere in this hitherto unpublished holograph material is there any textual evidence of the final paragraph of the Pythagoras chapter. The next manuscript leaf contains the opening passages of the Heraclitus chapter.